PSLQ

[AlexFekken]

I like the irony, or if you like: paradox, of these functions. So sorry, if you are not interested, for the caveat rant for those expecting silver bullets wherever they go.

The motivation for PSLQ (and QPI) seems to be that "arbitrary numbers" are likely to be of a special type: rational, semi-rational (in QPI sense), or algebraic (or in the same field extension of Q as some other numbers). This is also illustrated by the various naming suggestions for QPI, especially those involving the word "exact" or otherwise suggesting that you are (always, usually?) getting a "better" representation of a number. And in situations where that is actually the case these functions are of course very useful.

But, as a sanity check, from a mathematical perspective these special types of numbers have measure zero, i.e. the probability that a randomly picked real number (within a given range, say between 0 and 1) belongs to one of these special types is 0, rather than close to 1. So mathematically your expectation that these functions will be useful when working with arbirtrary numbers should be 0 too.
And at the other end of the scale, i.e. when you take into account that machine numbers are not "arbitrary (real) numbers" to begin with: all machine floats are already rational, so why bother about finding "better" (?) rational, semi-rational or algebraic representations? This is borderline pseudo-science, depending on the context.

And on top of this it is all relative as well. "3.141592653" coming out of a long calculation on the Prime may be "better" represented as pi, whereas coming out of a similar calculation using Free32 it is extremely unlikely that pi is a better representation (unless there is an issue with the calculation itself). Who is going to check whether the accuracy justifies or invalidates the "better" representation (not your average student, I would expect)?

So it seems that we are dealing with some shadowy area between Q and R, representing it all with just a subset of Q in a way that actually seems to make some sense. But only as long as you know what you are doing.